Do numbers “exist”? What about properties? I know my red apple exists, but does “redness” itself exist?
The existence of abstract objects seems, at first, like a deep metaphysical question. In fact, it’s a question of the pragmatics of language.
A couple of quick definitions. The view that abstract objects do exist is called “Platonism.” The view that they don’t is “nominalism.” Those who think they do exist, but only in the mind, are “conceptualists.”
Let’s take the case of numbers. A typical nominalist argument says that while you may bump into two apples somewhere along your travels, you’re never going to bump into “2.” There is no such thing independent of our descriptions of states of affairs. And that’s all abstract objects are – descriptions. They exist only in language.
The conceptualist replies: the fact that they exist even just as descriptions demonstrates that they do exist – in the mind. Abstract objects are mental fictions, and as such, they exist.
The Platonist’s retort: how do you explain that we all come up with the same mental fictions? When you and I speak of “the number of apples here,” we’re not talking about two different fictions that each of us came up with and which we happened to give the same name to. We’re speaking about the same thing: the number 2!
The Platonist may add that science corroborates the existence of numbers. Science predicts reality, and it does so through the use of numbers. This verifies the fact that numbers aren’t just some arbitrary or socially conditioned way of interpreting the world. They’re a part of how the world really works.
With whom should we agree? With all of them, of course! The nominalist is obviously correct in saying that we’re never going to bump into a “2” on the street, and that it’s only a description. The conceptualist is correct too: the idea of “2” is quite clear in my mind. And, radical skepticism aside, how can we deny the Platonist’s point? Of course we have the same thing in mind when we talk about “2.” The concept exists independently of any particular mind.
The three views aren’t in any substantive disagreement about the status of abstract objects. The Platonist concedes that we’ll never “bump into” a “2.” The nominalist doesn’t deny that I really have a thought about the number 2. The conceptualist may or may not agree with the Platonist on the point that when we talk about “2,” we’re talking about the same “thing.” But if he doesn’t, he’ll concede that we’re talking about the same type of thing, that the things we’re talking about have mostly the same properties, that there’s something we share when we converse. Unless he’s a radical skeptic, he does not deny the possibility of communication.
The only real difference between the views, then, is what counts as existence. Should we say that something that exists only in minds really exists? Does a concept that is shared by more than one person have a stronger claim to existence than something that exists only in one individual mind?
As I’ve written elsewhere, meanings are stipulations. “Exist” is just a word – we decide what it means. As with most words, we’ve decided its meaning gradually and messily, through use. For this reason, its meaning isn’t precise. Do abstract objects qualify under the definition of “existence”? The word’s meaning, as determined by our everyday use of it, doesn’t answer the question. It’s incomplete.
Once we understand this, a better question presents itself: what should the word mean? What would an optimally useful language say about the word “exist”?
There are two approaches I can see to this question. One says that language should be as robust as possible. The existence of the Eiffel Tower, of an image I currently have in my mind, and of the number 2 are all different kinds of existence. If our language were perfect, it would distinguish between these kinds. This would help avoid confusions, such as the longstanding debate over the ontology of abstract objects.
Another approach is one that aims to strip down ontology as much as possible. Let’s not say that something exists unless it is absolutely necessary. The question is, absolutely necessary for what? Bertrand Russell tried something like this. His absolute was sensory experience (at some point in his career – he often changed his mind). For Russell, sensory experience is all we can be sure exists. Anything that we need in order to understand and describe sense experience we must posit as existing.
W. V. O. Quine took a similar approach but used a different reference point. For Quine, what we can trust most is physics. That’s our most reliable and accurate field of rigorous study. Whatever is necessary to make sense of physics we must say exists. Quine began his career as a nominalist. Over time, he discovered he could not account for physics without set theory. He, therefore, granted existence to mathematical objects and became a Platonist.
Decades later, Hartry Field wrote Science without Numbers and showed that numbers are not necessary to account for physics. Physical theories could be described in purely logical terms, without recourse to mathematics. It would be incredibly cumbersome and inconvenient, but it could be done.
So, what approach would be the more useful to adopt, robust or minimalist? If minimalist, should we privilege sense experience or science? These are interesting questions that are not easy to answer. Understanding that they are pragmatic questions about language, however, at least gives us an entry point from which to tackle them.